NZ Scholarship Calculus study plan

Manipulate rational, radical and absolute-value expressions, including splitting rational functions into partial fractions with repeated linear and irreducible quadratic factors.

Prove statements by mathematical induction, covering divisibility, summation and inequality cases, and state the induction hypothesis explicitly every time.

Solve inequalities and sketch graphs of functions built from absolute values, roots and asymptotes, stating domain and range precisely.

Construct short proofs by contradiction and produce counterexamples to disprove false general statements.

Find inverse functions and analyse domains and ranges of composites, including composites involving restricted-domain trigonometric and inverse-trigonometric functions.

Differentiate from first principles, including trigonometric, reciprocal and root functions, showing the limit steps clearly.

Apply the chain, product and quotient rules fluently to deeply nested expressions without losing algebraic accuracy.

Set up and solve optimisation and related-rates problems, writing the constraint equation before differentiating.

Differentiate implicit and parametric relations, including finding second derivatives of parametric curves in terms of the parameter.

Analyse curves completely: stationary points, concavity, points of inflection, asymptotes and behaviour near excluded values, then produce an accurate sketch.

Use the derivative to prove inequalities and to establish monotonicity on a stated interval.

Differentiate exponential, logarithmic (including base a) and inverse trigonometric functions, and justify each result when asked.

Recognise standard integrals on sight and choose between substitution, completing the square and trigonometric substitution for unfamiliar integrands.

Integrate by parts, including repeated application and cyclic cases that require solving for the integral algebraically.

Integrate rational functions using partial fractions and handle improper cases where the numerator degree is not lower.

Evaluate definite integrals and exploit odd, even and periodic symmetry to shorten computations.

Compute areas between curves and volumes of revolution, including rotation about a line other than an axis and integration with respect to y.

Solve separable and first-order differential equations, and translate a written modelling situation into a differential equation with correct initial conditions.

Apply the Fundamental Theorem to differentiate integrals with variable limits, treating the integral as a defined function.

Evaluate improper integrals and determine convergence or divergence with justification.

Prove trigonometric identities, including those requiring the t = tan(x/2) substitution, sum-to-product forms and double-angle manipulation.

Solve trigonometric equations over a stated domain and write general solutions using the correct period.

Work with inverse trigonometric functions: state domains and ranges, differentiate them and integrate the resulting forms.

Apply compound-angle and product-to-sum identities to evaluate definite integrals and simplify expressions before differentiating.

Convert fluently between rectangular, polar and exponential forms, and compute powers and nth roots using De Moivre's theorem.

Solve equations of the form z^n = w and mark the roots on an Argand diagram, describing their symmetry.

Sketch and describe loci such as |z - a| = |z - b| and arg(z - a) = theta, and translate between algebraic and geometric descriptions.

Use complex numbers to derive multiple-angle identities and to factorise real polynomials into linear and irreducible quadratic factors.

Sit two full-length three-hour simulations under exam conditions, calculator-free, with strict timing and no notes.

Classify every error from both papers into algebra slips, technique gaps, misread questions and time-management losses, and rank them by frequency.

Re-solve every missed question from scratch after a 48-hour gap and confirm each is now correct without prompts.

Reconstruct a one-page formula and technique sheet from memory and check it against the week 3 sheet.